INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY. VOL. X, 811-835 (1976)
A Comparison of Orthogonalized Plane
Wave and Augmented Plane Wave
Methods for Calculating Photodetachment
Cross-Sections
MANIJEH MOHRAZ AND LAWRENCE L. LOHR, JR.
Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, USA
Abstracts
The orthogonalized plane wave (OPW) method of calculating electronic continuum wavefunctionsis tested by the computation of photodetachment cross-sections and angular distributionsfor
gaseous halide anions. The results are compared to those obtained by a related augmented plane wave
(APW) method involving the exact solution of a single-particle Schriidinger equation containing a
piecewise Coulombic potential energy. These comparisons, as well as other involving experimental
and theoretical cross-sections from the literature, indicate that OPW cross-sections are, at best, only
semi-quantitatively reliable for describing photodetachment even at low photon energies, and that
OPW cross-sections should be calculated using the dipole length operator rather than the dipole
velocity operator.
La mtthode des ondes planes orthogonalistes (OPW) pour calculer des fonctionsd’onde du spectre
continu a Ct6 test6 par un calcul des sections efficaces et des distributions angulaires assocites a des
processus de photodttachement pour des anions halogtno’ides. Les rksultats sont comparts a ceux
obtenus par la mtthode des ondes planes augmenttes (APW), qui comporte la solution exacte d’une
tquation de Schrodinger B une particule, avec un potentiel coulombien par morceaux. Ces comparaisons ainsi que d’autres bastes sur des sections efficaces exptrimentales et thtoriques de la
litttrature indiquent que les sectionsefficaces OPW ne sont guerre dignes de confiance m&mepour des
tnergies photoniques basses, et qu’elles doivent &re calculies avec I’optrateur dipolaire de longueur
plutdt qu’avec celui de vitesse.
Die OPW-Methode (orthogonalisierte ebene Wellen) fur die Berechnung von elektronischen
Kontinuumfunktionen wird rnit einer Berechnung der mit Photoauslosevorgangen
zusammenhangenden Querschnitte und Winkelverteilungen fur Halogenidanionen getestet. Die
Ergebnisse werden mit denen verglichen, die durch eine APW-Methode erhalten werden. In dieser
Methode wird eine exakte Lasung einer Schrodingergleichungfur ein Teilchen rnit einem stuckweise
coulornbischen Potential erhalten. Diese Vergleichungen als auch andere, basierend auf experirnentelle und theoretische Literaturquerschnitte, deuten an dass die OPW-Querschnitte nicht einmal fur
geringe Photonenenergien zuverlassig sind, und dass sie mit dem Dipollange-operator eher als mit
dem Dipolgeschwindigkeitsoperatorberechnet werden sollen.
1. Introduction
The rapidly increasing use of photoelectron spectroscopy has stimulated much
recent interest in the calculation not only of ionization potentials but also of
photoionization cross-sections and angular distributions. The desire to treat
theoretically molecules of chemical interest has led to many studies using either
plane waves (PW’s) or orthogonalized plane waves (OPWs) to represent the
ejected electrons [1-1 11.The use of OPW continuum functions may be viewed [4]
811
@ 1976 by John Wiley & Sons, Inc.
812
MOHRAZ AND LOHR
as the use of plane wave (PW) continuum functions together with considerations
of the nonorthogonality of PW’s t o core orbitals and of the requirement of total
wave-function antisymmetry. The results obtained to date by this method for a
variety of small molecules appear to be semi-quantitatively reliable and very
useful to photoelectron spectroscopists in making assignments. Particularly valuable have been the studies [10,111 showing how the variation with photon energy
of the cross section for ionization from a given molecular orbital is related to the
symmetry and composition of that orbital. However, serious misgivings have
frequently been raised [12] about even the approximate validity of the OPW
method. It is the object of this paper to explore the relationship of the OPW
method to more nearly exact methods of calculating continuum wave-functions
and to present numerical results for atomic photodetachment cross-sections and
angular distributions.
We have chosen to study photodetachment of anions rather than photoionization of neutrals because the continuum wave-functions in the former case are
phase-shifted plane waves, which are more closely related to OPW’s than are the
phase-shifted Coulomb waves appropriate to the latter case. The specific systems
studied are the gaseous halide ions, for which there have been numerous experimental and theoretical investigations of photodetachment cross-sections. Since
these studies are largely confined to an energy range not exceeding 1Ry (13.6 eV)
above the detachment threshold, our studies are made in the range from zero to
10 Ry to facilitate comparisons. Thus, we are not considering in detail the high
energy range for which the PW method and its derivative, the OPW method, are
presumed to be most nearly valid. The compensating factor, as mentioned above,
is the choice of anions rather than neutrals to study, thus eliminating the need to
consider a long range Coulomb interaction. However, the calculation of cross
sections for p-electron detachment, as in the case of the halides, is a demanding
test since there are two final states for the electron, an s-state with a comparatively
large phase shift and a d-state with a comparatively small phase shift. This study
serves as a model for the detachment of non-bonding electrons from such species
as OH-, SH-, SeH-, NH;, PH;, and ASH;, each isoelectronic with a halide ion
and each studied by either crossed-beam [13] or ion cyclotron resonance [14]
techniques. Earlier PW calculations of photodetachment cross sections include
the studies of Bates and Massey [ 151 and Chandraskhar [16] on H-, and those of
Moskvin [17, 181 on Li-, Na-, K-, C-, N-, 0-, F-, and C1-. A recent study by
Reed et al. [ 191reports cross-sections for H-, C - , 0-,F-, OH-, O,, CN-, C,, and
C5H; calculated using PW’s orthogonalized only to the orbital from which
detachment occurs. Other theoretical methods are described at the end of the next
section.
2. Theoretical Methods
The differential cross-section for producing photoelectrons in the solid angle
dfl is given [l, 3, 91 in the dipole approximation by the expression
CALCULATING PHOTODETACHMENT CROSS-SE(JT1ONS
813
where e is the magnitude of the electron charge, m the electron mass, c the
velocity of light, w the circular frequency, u a unit vector in the direction of the
electric polarization, a and b the initial and final states respectively, pn the linear
momentum operator -ihV, for the nth electron, and p ( E ) the density of final
states with an electron kinetic energy E = hw -BE, where the binding energy BE
for photodetachment equals the electron affinity EA. If the initial state (a1 is a
determinant of doubly occupied atomic orbitals and the final state ( 6 )is a spin
singlet constructed by promoting one electron from the jth atomic orbital to an
unbound orbital ( k ) ,with no change in the shape of the other atomicorbitals, then
(aIC~~lb)=2’’~(jlp(k)
(2)
It
where the factor 2l” accounts for the fact that there are two electrons in orbital j .
The matrix element of the linear momentum operator p in Eq. (2) may be written
in terms of dipole length operator r as
where Ek -Ej corresponds to the photon energy hw.
In this study each bound atomic orbital (jl is taken to be a superposition of
Slater-type basis functions. The orbitals used are those obtained by selfconsistent-field (SCF) calculations on the atomic halides [20-231.
Two types of continuum state functions ( k )are used in our photodetachment
cross section, namely orthogonalized plane waves (OPWs) and wave-functions
derived from a central-field model. The latter type of function is related to a plane
wave which is phase-shifted at distances far from the nucleus and which is
augmented by Coulomb-type functions at distances close to the nucleus. Therefore it will be referred to as an augmented plane wave (APW).
The adaptation of Eq. (1) to OPW final states has been given by several
authors [4, 91 and will be only briefly described. A box-normalized plane wave
(PW)
Ik) = (PW(k))= LP3” elk.‘
(4)
where L is the box length, has a corresponding density of states
so that Eq. (1)reduces to
where the subscript j on d u / d i l denotes the atomic orbital from which detachment occurs. The magnitude of k in Eqs. ( 5 )and ( 6 )is given by
814
MOHRAZ AND LOHR
where EAj is the electron affinity for the j t h orbital. Since the PW is an
eigenfunction of the linear momentum p with eigenvalue hk,
Using the dipole length formulation,
The integrals in the two expressions are quite different. In Eq. (8)(jlPW(k)) is the
overlap of the initial bound state with the PW, or simply the Fourier transform of
the bound state into momentum space. In Eq. (9) the integral (jlrlPW(k))
corresponds to a transition moment from the initial states (jl to the final state
(PW(k)). In the spirt of Eq. (8) one could write the matrix element in Eq. (9) as
(rj(PW(k)), which is the overlap of a state (rjl with the PW.
Schmidt orthogonalization of the PW to the bound state yields
lOPW(k)) =“IPW(k))
-C1 (@‘W(k))lI)I
(10)
where the sum is over all occupied orbitals / I ) . The normalization constant N is
given by
N = [ l - C (IJPw)(PwIl>]-”2
1
As the size of the box within which the P W s are normalized becomes infinite, the
overlap (IIPW) becomes negligible compared to unity and N approaches unity.
Substituting Eq. (10) in Eq. (6),
The use of OPW’s introduces into the cross-section expression bound-to-bound
dipole matrix elements of the form (jlpll) multiplied by monopole matrix elements (f(PW(k)).
The differential cross-sections are calculated for two fixed orientations of the
polarization u with respect to the electron momentum hk, namely for the
polarization u perpendicular to hk and for u parallel to hk. For detachment from
orbitals other than s orbitals an averaging procedure is required and may be
carried out in either of two ways: a) by specifying the type of the orbital (for
example px,p,,, or pz in the case of detachment from a p orbital) and then averaging
over the orientation of its axes with respect to the laboratory axes; or b) by
calculating a simple average of the expressions for detachment from all the
different orbitals in a given subshell for a fixed direction of u relative to k.
In summary, the integrals that have to be evaluated in the OPW procedure are
of three main types: overlap integrals (jlPW) of atomic wave-functions with a PW,
CALCULATING PHOTODETACHMENT CROSS-SECTIONS
815
bound-to-bound transition integrals (jlrlf) and (jlpll), and dipole transition
integrals of the form (j(r1PW).
In the second, or APW, method the continuum state wave-function is obtained
by solving the radial Schrodinger equation
where 1 is the angular quantum number of the photoelectron, V,,I,(r)(which is
designated with the subscript nl' to indicate that detachment is from the nt'th
subshell) is an effective potential energy, and PEI(r)= rREl(r) is the continuum
radial function. We have chosen to use a modification of the procedure developed
by McGuire [24] and extended by Chapman and Lohr [ 2 5 ] for the solution of Eq.
(13). In this procedure V ( r ) is approximated by a model potential energy for
which an exact solution of the radial equation can be determined. The potential
energy corresponds to an electron in the field of a positive charge Ze at the origin
surrounded by a series of concentric shells at Ri, each having a uniform distribution of negative charge -2,e. The potential energy is piecewise Coulombic in all
regions except the outermost, and hence solutions can be expressed in terms of
Whittaker functions. These functions of the dimensionless variable kr are
evaluated in terms of their power series expansions for kr 5 11and with asymptotic formulas for kr > 11. They involve a number of infinite but convergent series
which are evaluated by approximating them as finite sums.
For the outermost region where V , ( r ) is zero, the solutions are spherical
Bessel and Neumann functions. It is the form of the solutions in this outermost
region that distinguishes the procedure from that used by McGuire [24] and by
Chapman and Lohr [25] for describing the photoionization of neutral atoms. For
each region of potential a general solution is constructed from the two particular
solutions required to form the general solution of the second-order differential
equation (13). The coefficients are evaluated by the application of suitable
boundary conditions. A solution for the entire range of potential is then determined by insuring that the wave-function and its derivative are continuous at each
boundary of the potential and that the wave-function has an appropriate form at
limits of small and large r. The limiting boundary conditions for the continuum
wave-function at small and large r are
as r+O
PEI= rREI(r)+ 0
and
PEI( r ) = rREl( I ) + k
-'sin
where is the phase shift of the lth partial wave relative to a free wave (for which
V ( r )= 0 for all r ) due to the attractive potential of the residual atom. The phase
shift is related to coefficients A and B for the outermost (nth) region by
tan 6 = - B / A
(15)
816
MOHRAZ AND LOHR
The phase shift is assigned a positive sign to indicate that the potential is attractive.
Since the spherical Bessel functions of order 1 which occur in the outermost region
may be viewed as spherical projections of a plane wave, the total continuum wave
may be viewed as the lth component of a phase-shifted plane wave suitably
augmented by Coulomb waves in the inner regions, and hence its designation as an
“augmented plane wave” (APW).
In order to characterize the piecewise potential energy V ( r ) , the set of
parameters Riand Zimust be specified. This is done by fitting [25]the piecewise
potential energy to a reference potential energy Vnl,(r),
which is obtained from
SCF orbitals for the core atom and which has the form
-Ze2
Vnl,(r)= r
e2
p ( r ’ ) dr‘+ e 2
I“
in which
The summation is over the n - 1 electrons in the residual atom. The occupation
number w ~ , for
, the E.LAth orbital includes electrons of both spins, but the exchange
interaction between the residual atom and the continuum electron is not included
explicitly. Note that the summation in Eq. (17) does not include the electron which
is being ionized, so that the so-called “self-Coulomb energies” are absent.
Parameters for the model potential energy are then determined by fitting r times
the reference potential energy of Eq. (16) with a series of line segments. The
charge at the origin is taken as the true nuclear charge Ze, and the sum of the shell
charges is -Ze. Thus, the model potential energy approches the correct asyrnptotic limit of zero. Since the fitting requires that the model and reference potential
energies be equal at each of the boundaries, the model potential energy is more
negative than the reference potential energy between boundaries. Exceptions to
this occur for radii greater than or just slightly less than the radius of the outermost
sphere.
If the bound-state orbitals in Eq. (17) are taken as those for the negative ion
before detachment, then the detachment is assumed to be electronically vertical.
This specifically neglects effects due to the relaxation of the core atom. The
reference potential energy can also be calculated using bound-state orbitals of the
residual neutral atom, in which case the potential energy corresponds to the
relaxed core.
In an attempt to account for the polarization of the core, a polarization term
may be added to the reference potential energy of Eq. (16). This term is taken as
Vp(r)= -
a eL
2(r2+ r;)’
where a is the polarizability and rp corresponds roughly to the atomic core radius.
It should be noted that the polarization term of Eq. (18) is not included explicitly
in the model potential. The continuum wave-functions for all n - 1 regions are still
calculated for a piecewise Coulombic potential energy. The only effect of polari-
CALCULATING PHOTODETACHMENT CROSS-SECTIONS
817
zation is to make the reference potential energy to which the model potential
energy is fitted more attractive.
The angular distribution of ejected electrons is characterized by an asymmetry
parameter p which appears in the differential cross-section [26]for polarized light
as
du
dln
-
u
[ I +~P,(COS
47r
el]
where 19 is the angle between k and u and P2(x)= ( 3 x 2 - 1)/2. The total subshell
cross-section u is the intensity parameter reported in Section 3. In the OPW
method p is conveniently obtained from
p = 2 ( u ~ ~ - u d / +2u,)=
(u~~ 8 d u ~ ~ - ~ * ) / 3 ~
(20)
where UIIand u,.denote the values of the differential cross-section for kllu and
k lu,respectively. In the APW method, as with other central field methods, @ for
p electron detachment or ionization is given by the Cooper-Zare [26]relationship
where R, and Rd are the radial transition moments to the s and d final states and
where @ is the difference between the s and d phase shifts.
Other theoretical methods have been used for calculating photo-detachment
cross-sections. Several of these are described below to show their relationship to
the present work. The main difference among the methods is in the type of the
bound and continuum state wave-functions that are employed and the manner in
which they are obtained. The notations Pnl(r)and PEI(r)are used to designate the
bound and the continuum radial wave-function, rRnI(r)
and r&(r), respectively.
Cooper and Martin [27] calculated photodetachment cross-sections for 0-,
C-, C1-, and F-, making the assumption that Pnl(r)and PEl(r)are eigenfunctions
of the same one-electron Hamiltonian having a potential energy
where the first two terms correspond to the electrostatic potential energy. The
factor p(r’)is the electronic charge distribution derived from the available Hartree
or Hartree-Fock wave-functions. The correction term (with a) is allowed to
absorb the effects of polarization and exchange as they effect the binding energy
Eb,which is taken as an experimental electron affinity. The factor rp is taken to be
the average core radius. The bound function Pnr(r)is then obtained treating a as
an eigenvalue. Using the calculated value of a, the continuum-state wavefunctions PEI(r)are then evaluated for various values of electron energy.
The chief advantages of this procedure are that it yields bound radial orbitals
with correct binding energies and that it includes polarization effects, with the
potential energy approaching the correct form -a e2/2r4 for large r. A major
818
MOHRAZ AND LOHR
shortcoming is that the semi-emperical parameters (Y and rp do not arise naturally
from the formalism and are introduced somewhat arbitrarily.
Using a related procedure, Robinson and Geltman [28] calculated photodetachment cross-sections for a number of atomic negative ions. The potential
energy which they used has the form
where VHs(r)is the Hartree-Fock-Slater potential energy for the neutral atoms,
evaluated by the methods of Herman and Skillman [ 2 9 ] . The second term
removes the Coulomb tail of VHs and the third term introduces the effects of
polarization. As in the previous case, the parameter rp is chosen somewhat
arbitrarily to correspond to core radius. The atomic polarizability (Y is chosen from
the best existing theoretical or experimental values, while the parameter ro is
chosen such that the resulting potential will support an np bound state with an
assumed binding energy. The bound- and continuum-state wave-functions are
then calculated in a manner similar to the method of Cooper and Martin.
Garrett and Jackson [ 3 0 ] used a more sophisticated procedure to calculate
photodetachment cross sections of 0-.The bound-state wave-functions P,,(r) are
calculated by the modification of the Hartree-Fock-Slater (HFS) method, in which
the potential energy is the sum of the electrostatic energy and an exchange energy
Vex(r)given in terms of the electron density p ( r ) by
where the coefficient A is varied until an eigenvalue of the 2 p electron is obtained
which corresponds to the experimental binding energy of 0-.This method of the
calculation of the bound-state wave-functions is different from that used by
Cooper and Martin or Robinson and Geltman in that a self-consistent calculation
is made each time the parameter A is adjusted rather than taking a single equation
and adjusting a parameter in the total potential function (the parameter rp in the
method of Cooper and Martin or the parameter ro in the method of Robinson and
Geltman) in order to match the desired eigenvalue. The continuum functions are
then obtained by numerical integration of the radial Schrodinger equation
containing a polarization potential obtained from an application of first-order
perturbation theory to the Hartree-Fock (HF) atomic system. The method leads to
substantial improvement in the agreement of theory and experiment over the
methods of Cooper and Martin or Robinson and Geltman. The major difference
in their results for 0- as compared to those of the previous methods stems from
differences in the continuum functions which are very sensitive to small differences in the polarization potential.
A somewhat different approach is the close-coupling method used by Conneely, Smith, and Lipsky [ 3 11for electron impact and photoionization as well as
CALCULATING PHOTODETACHMENT CROSS-SECTIONS
819
for photodetachment. In lieu of a polarization potential their method explicitly
includes low-lying excited terms in the final-state wave-functions, although again
SCF initial-state functions are chosen. Very recently, Ishihara and Foster [32]
studied the photodetachment of C- and F- using many-body perturbation theory
in which they included the effects due to correlation and to intrachannel interactions. The calculated cross-sections are in very good agreement with the experimental results, showing the importance of the above corrections in the crosssection calculations.
3. Results and Discussions
A . General Remarks
The photodetachment cross-section u and the asymmetry parameter p are
calculated by the OPW and APW methods for the np photodetachment of halide
ions. In all cases both the dipole length and dipole velocity formulations are
employed. In order to facilitate comparisons the same SCF bouhd-state functions
are used in the PW, OPW, and APW calculations. These functions enter the OPW
calculations both as the initial-state functions ( j ( and as the functions to which a
PW is Schmidt orthogonalized. They enter the APW calculation both as the
initial-state functions (jl and as the functions used to calculate the reference
potential energy in Eq. (16). Since SCF functions for X- rather than Xo are used
throughout, effects associated with core relaxation are ignored. The spin-orbit
splitting of the np5 core of the Xo atom is ignored except in the series of APW
calculations including core polarization, where the splitting is considered in order
to facilitate comparisons of our results with other calculated values and with
experimental values.
Table I is a list of the electron affinities for the halide ions. The values in the
first column are one-electron energies [20-231 corresponding to the bound-state
orbitals of X-. The values in the second column correspond to the difference in the
neutral and negative ion total energies, where Goth energies are calculated
[ZO-221 from wave-functions with comparable basis sets. The third column
corresponds to the experimental electron affinities [33]. For most of the ions the
TABLE1. Electron affinities(eV).
Atom
One-Electron
Energies
D i f f e r e n c e of
Total E n e r g i e s
Experimentale
F
4 . 93a
1.39
3.45
c1
4.13b
2.59
3.61
Br
3.7Jc
2.58
3.36
3.10d
__
3.06
I
a
Ref. [20].
Ref. [21].
'Ref. [22].
Ref. [23].
Ref. [33].
820
MOHRAZ AND LOHR
three electron affinities listed in Table I are significantly different. This poses a
problem in choosing among them. Since for negative ions the calculated energies
are not usually very accurate, the experimental electron affinities provide a better
description of the process. On the other hand, the theoretical one-electron
energies correspond to the bound-state functions which are used to calculate the
photodetachment cross-sections, and in that sense they might be more appropriate.
In the cases where the values are different, the choice of one or the other
makes a significant difference in the relative magnitude of the cross-sections
calculated by the dipole length and dipole velocity methods, especially at very low
electron energies. This is easily illustrated if one looks at the two expressions
ffL
ffV
- ( E +E A ) ( M E ~ ( ~
(24)
- ( E+1EA) W”I2
where the subscripts L and V correspond to dipole length and dipole velocity
formulations, respectively. EA is the absolute value of the electron affinity and E
the kinetic energy of the photoelectron. In Eq. (24) the cross-section is directly
proportional to the photon energy hw = E + EA, whereas in the formulation of
Eq. (25) it is inversely proportional to this factor. For example, at energies close to
threshold the cross-section uLfor F- increases by about 40% if the one-electron
energy is used, while the cross-section uv decreases by the same amount.
Therefore, at low energies the calculated cross-sections are very sensitive to the
value used for the electron affinity. For the calculations reported here the
experimental electron affinities are used.
B. A Comparison of PW and OPW Results
The variation of the PW and OPW cross-sections with photoelectron energy
for ions F-, C1-, Br-, and I- are shown in Figures 1through 4, respectively. For F(Fig. 1) with the dipole length formulation, both the magnitude and th’e shape of
the cross-section are significantly different for the PW and OPW results in the
energy range from 0 to about 2 Ry. The two approach each other as one moves to
higher energies. The same is also true for the dipole velocity formulation. There is
a similar pattern for the other three ions (Figs. 2-4). There is a similar pattern for
the other three ions (Figs. 2-4). The difference is even greater if one compares the
results of the two formulations in each of the two PW and OPW methods. As
mentioned earlier, one must be cautious when comparing the cross-sections
calculated by the dipole length and dipole velocity formulations, since at low
photoelectron energies the relative magnitude of the two is very sensitive to the
choice of the electron affinity. Indeed, using the theoretical orbital energy gives a
better agreement between the two formulations at electron energies below
0.05 Ry. However, as the electron energy increases, the use of a greater electron
affinity makes the agreement worse as the cross-sections calculated with the
82 I
CALCULATING PHOTODETACHMENT CROSS-SECIIONS
1
F-
K
E (Ry)
Figure 1. Cross-section (J or in Mb. vs. electron kinetic energy in Ry for F-. OPW
OPW (velocity) by (--),
(length) results denoted by (- . - .), PW (length) by (-4,
and PW (velocity) by (- - - -). Curves in Figures 1-8 and 10-12 are drawn through
points computed at intervals of 0.1 Ry between zero and 1.0 Ry and at intervals of
1.O Ry above 1.O Ry. Finer meshes were employed in some cases.
c1-
I
0
2
6
4
KE
0
I
(Ry)
Figure 2. Cross section cr in Mb vs. electron kinetic energy in Ry for C1-. Curves
labeled as in Figure 1.
dipole length formulation are becoming greater than those calculated with the
dipole velocity operator. The “dips” in the plots of (+ at low electron energies are
due to the fact that the cross-section for the photodetachment from a p orbital is
the sum of the cross-sections for the transition to the s and d continuum waves.
For the s transition the cross-section rises rapidly at very low electron energies,
reaches a maximum, and then drops. For the d transition the cross-section rises
gradually with energy and reaches a maximum at much higher energies.
An important result that is not readily apparent in Figures 1-4 is that all of the
curves except for those obtained using PW’s and the velocity operator have the
822
MOHRAZ AND LOHR
Figure 3 . Cross-section u in Mb vs. electron kinetic energy in Ry for Br-. Curves
labeled as in Figure I .
K E (Ry)
Figure 4. Cross-section u in Mb vs. electron kinetic energy in Ry for I-. Curves
labeled as in Figure 1.
-
threshold behavior (+ k expected from Wigner’s rules [34] for the detachment of
a p-electron. By contrast, the curves for the PW results with the velocity operator
have the threshold form u k3’* for the detachment of any electron, a result that
is only correct for s- or d-electron detachment. The conclusion is that use of the
length operator is necessary if P W s are used in order to have the correct threshold
behavior. The same conclusion was reached by Reed et al. [ 191in studies involving
orthogonalization of the PW only to the orbital from which detachment occurs.
However, the additional orthogonalization of the PW to s orbitals builds into
expression (14) the requisite final states for IT so that either operator yields the
-
823
CALCULATING PHOTODETACHMENT CROSS-SECTIONS
correct threshold behavior. By contrast, at the higher energies shown in Figures
1-4 the cr values are more dependent on the choice of operator than on
orthogonalization, with the length values lying significantly above the velocity
values for all halides.
Figures 5-8 show the variation of the PW and OPW values of the asymmetry
parameter p with photoelectron energy. The PW-dipole velocity formulation p
equals 2 at all electron energies and is not shown. For the other three methods, 0
starts at zero, implying a spherical distribution at threshold. It drops to a minimum
of - 1 at about 0.1 to 0.2 Ry and then rises to a value close to +2. The shape of p is
approximately the same for all three methods. Also shown in these figures are
P
FI
0
2
a
6
4
10
K E (Rgj
Figure 5. Asymmetry parameter p vs. electron kinetic energy in Ry for F-. OPW
OPW (velocity) by (--),
(length) results denoted by (- . - .), PW (length) by (-),
and APW (length) by (-).The
PW (velocity) value ofp is 2 at all energies.
2-
--
.----
\
'.
/--
c1-
\
\
\
0
2
4
6
I
8
10
K E (Rg)
Figure 6 , Asymmetry parameter p vs. electron kinetic energy in Ry for C1-. Curves
labeled as in Figure 5.
824
MOHRAZ A N D LOHR
K E (Ryf
Figure 7. Asymmetry parameter in 0 vs. electron kinetic energy in Ry for BrCurves labeled as in Figure 5.
0
2
6
4
KE
a
10
(Ry)
Figure 8. Asymmetry parameter 0 vs. electron kinetic energy in Ry for I-. Curves
labeled as in Figure 5.
APW-dipole length p values which are discussed in Section 3D. For C1-, Br-, and
I-, after reaching a maximum p does not level off or decrease gradually, as for F(Fig. 5), but sharply drops to a minimum and then rises again. In the OPWdipole length method of Figure 6 for C1- the second minimum occurs after 10 Ry.
Going from C1- to I- this minimum is shifted towards lower energies. For Br- it
occurs at about 7.4 Ry and for I- around 4.6 Ry. Although the OPW p values are
not calculated from Eq. (21), the equation may be used to analyze the results.
Since the OPW phase shifts are zero, p equals zero at energies for which Rd = 0 or
Rd = 2Rs ; any relationship between the number of such crossings, and hence
oscillations in 0, to the number of bound-state radial nodes requires further
exploration, although there is no relationship in the corresponding pliotoionization of np electrons from neutral rare gas atoms [35, 361. It has recently been
CALCULATING PHOTODETACHMENT CROSS-SECTIONS
825
stated [37] that since p is so crucially dependent on the phase shifts there is no
possibility, except by accident, of p’s obtained by PW methods, and presumably
by OPW methods, being correct. While it would certainly be unjustified to attach
other than semi-quantitative significance to the results in Figures 5-8, we do note
that the curves have the correct threshold behavior for p-electron detachment as
found theoretically by Cooper and Zare [38] using the Robinson-Geltman
potential [28] and as found experimentally by Hall and Siege1[39]. Specifically,p
is zero at threshold, falls rapidly to a value near -1, and then rises through zero to
a value near 2.
C. A Comparison of APW and OPW Results
In this section the APW cross-sections for the photodetachment from the p
orbital of halides are compared with the OPW results of the preceding section.
Spin-orbit coupling is again neglected so that the cross-sections represent the
sum over the
levels of the residual atom. Cross-sections were
and
computed for all halides, but are listed for F- only in Table I1 with E denoting the
photoelectron energy and subscripts L and V the dipole length and dipole velocity
formulations, respectively. The OPW results are the same as those shown in
Figures 1-4.
TABLE11. Photodetachmentcross-sections (Mb) for F-.
0.0
0.0
6.9
6.8
14.0
8.5
6.6
12.4
0.0
0.0
2574
0.1
4.5
2007
0.2
6.0
1645
0.3
7.2
9.7
6.7
11.1
13 93
0.4
8.3
10.6
7.0
10.2
1209
0.5
9.2
11.3
7.4
9.6
3588
0.0
1067
0.6
10.0
11.8
7.8
9.2
955
0.7
10.8
12.1
8.3
8.9
a
864
0.8
11.3
12.2
8.8
8.7
727
1.0
12.3
12.2
9.6
8.2
404
2.0
11.9
9.8
10.8
6.0
214
4.0
6.5
5.2
7.6
3.3
146
6.0
3.4
2.7
5.3
2.0
110
8.0
1.9
1.5
3.7
1.3
89
10.0
1.1
0.9
2.7
0.9
Using experimental EA = 0.254 Ry.
826
M O H R A Z A N D LOHR
The results of the dipole length and dipole velocity methods are generally
different in magnitude. This may seem surprising since the APW continuum
function represents an exact solution of the Schrodinger equation with the model
potential energy V M ( r )However,
.
the SCF bound-state orbital is a solution of the
Schrodinger equation with a different Hamiltonian, so that the identity in Eq. (3)
does not hold. This point has been recently discussed by Starace [40] and by
Cohen and McEachran [41]. In addition, we chose experimental rather than
theoretical E A values, thus giving up any requirement of agreement of length and
velocity cross sections. For F- (Table 11) the agreement between the length and
velocity formulations is fair, and in most cases the difference is less than 25%. In
both cases the maximum cross section occurs between 1.0 and 2.0 Ry.
For all the halides there is a general agreement in the shape of the OPWL
cross-section and the position of its maximum with the APW results. However,
the magnitude of the OPWLmaximum cross-sections are generally lower than the
corresponding APW values. The OPW, cross-sections are very different from the
APW values and, as in the case of the length operator, the maximum OPW
cross-section occurs at lower electron energies.
In Figure 9 we illustrate the PW, OPW, and APW s continuum wave-functions
for F- at an energy of 1.0 Ry. The functions are plotted as P ( r )= rR(r),with an
I
r (o.u.)
Figure 9. The s continuum waves r R ( r ) vs. r in a.u. at an electron kinetic energy of
l.ORyforF-.APWdenotedby(-. -),OPWby(- -),andPWby(-).
arbitrary unit amplitude so that the curve labeled PW is simply krjo(kr)= sin (kr),
which is kr times the s component of the PW. The dashed vertical line at r = 5 a.u.
denotes the radius beyond which V,(r) is zero. We note that this is also the
approximate radius beyond which the PW and OPW curves coincide. For r <
5 a.u. the OPW function has one more maximum than does the PW function, but
there is of course no long range phase shift. Another feature of the OPW function
is that its nodes do not coincide with its points of inflection, so that there are
CALCULATING PHOTODETACHMENT CROSS-SECTIONS
827
singularities at the nodes in the local kinetic energy T (local) = (T'P)/q, where T
is the kinetic energy operator. Conversely, the OPW function may be viewed as an
exact solution of a Hamiltonian containing a singular potential E - T (local). By
contrast, the APW function is an exact solution of a Hamiltonian containing a
nonsingular potential, so we observe in Figure 9 that its nodes and points of
inflection coincide. The s phase shift at this energy is 1 . 5 7 ~or
, approximately
37~/2,so that the extrema at large r nearly coincide with the nodes of the PW or
OPW functions.
In Figure 10 we illustrate the PW and APW d continuum functions for F-,
again at an energy of 1.0 Ry. The PW function is simply krj2(kr)and may be used
Figure 10. The d continuum waves rR ( r )vs. I in a.u. at an electron kinetic energy of
1 .0 Ry for F-. APW denoted by (- . -) and OPW or PW denoted by (- -).
as an OPW function since F- has no occupied d orbitals. As expected, the APW
phase shift is quite small, namely 0 . 0 2 7 ~so
, that the PW and APW functions are
essentially identical at this energy.
D. Inclusion of Core Polarization and Comparisons with Other Results
APW continuum wave-functions are now evaluated for all halides using a
reference potential energy which includes an empirical core polarization term
V p ( r )of the form given in Eq. (18). The halogen polarizabilities are chosen either
from theory [42] (F) or experiment [28] (Cl, Br, and I) are given in Table I11
together with values of the parameter rp chosen to correspond roughly to the
atomic core radius. The effect of V J r ) on the parameters defining the model
potential energy V M ( r )is quite large. For C1- the effect is to make V M ( r )at
r = 3 a.u. about twice as attractive as in the case with no V p ( r )in the reference
potential energy. In addition, the region with V,(r) = 0 begins at r = 13.9 a.u.
when polarization is included in V, whereas the corresponding region begins at
828
M O H R A Z A N D LOHR
6.2 a.u. when polarization is not included. The polarization effects on VM(r)are
even greater for Br- and I-. As is well known from earlier studies [28], the
inclusion of polarization increases the maximum value of the cross-section and
moves its position to lower electron energies. For C1- the APWL maximum
without polarization is 46 Mb at a kinetic energy of 0.7 Ry, while the AP.WL
maximum with polarization is 57 Mb at 0.6 Ry.
Figure 11compares the APWL with polarization for F- to the OPWLresults, to
the theoretical results of Cooper and Martin [27], Robinson and Geltman [28],
0
0
I
I
0.1
0.2
1
1
I
0.4
KE (Ry)
I
0.6
I
1
I
0.8
I.
Figure 11. Cross-section c in Mb vs. electron kinetic energy in Ry for F-. APW
(length) results with polarization denoted by (- - -) and OPW (length) results by
(-.-.). Other results are those of Cooper and Martin [27] denoted by (.. ..),
Robinson and Geltman [28] by (-),
Ishihara and Foster [32] by (- -), and Mandl
[43] by the points @.
and Ishihara and Foster [32], and to the shock tube measurements of Mandl [43].
Except for electron energies below 0.2 Ry the OPWLvalues are much lower than
the others. The excellent agreement between the APWL calculations and Mandl’s
measurements, which have estimated uncertainties of *25%, must in part be
fortuitous, as our calculations ignore correlation in the initial state and both
correlation and exchange in the final state. However, our reference potential
energy used in calculating the APW final state does implicitly contain some
correlation and exchange effects uiu the polarization term. Mandl [43] also
compared his measurements to Robinson and Geltman’s calculations, as well as to
Moskvin’s PW calculations [181and to Berry and Reimann’s [33] and Popp’s [44]
near threshold measurements. As Mandl’s Figure 4 and our Figure 11 indicate,
829
CALCULATING PHOTODETACHMENT CROSS-SECTIONS
the Robinson-Geltman calculations agree reasonably well with experiment if
their values are scaled by approximately 0.7.
The experimental [45] spin-orbit splittings for the neutral halogens are given
in Table 111.We now include the effect of this splitting on the cross-sections for C1,
Br, and I by simply apportioning two-thirds of the computed cross-section to the
lower J = 3/2 level and one-third to the upper J = 1/2 level, being careful to use u
values computed for the desired kinetic energy, which for a given photon energy is
different for the two levels, and to correct the photon frequency factors in Eqs.
(24) and (25).
Figure 12 compares the APWL results with polarization for C1- to the OPWL
and other theoretical results. The APWv values, not'shown, are typically about
TABLE111. Polarization and spin-orbit parameters.
Atm
o
F
4 . 05''
(rlu
rp (nu)'
AEso ( R y ) =
1.5
0.0037
C1
2 3 . 5''
2.5
0.0 0 8 0
Br
24. g h
3.5
0.0336
1
40. g h
4.5
0.0693
-
Ref. [42].
Ref. [28]. Values chosen to be larger than a for the neighboring rare gases.
'Ref. [45].
dConverta tocrn3andrptocmforuseinEq.(18).
a
1/3 less than the APWL values over this energy range. The experimental u value
of Berry and Reimann [33], Muck and Popp [46], and Rothe [47] are not shown as
they are confined to a narrow range of energies not exceeding 0.04 Ry above
threshold (their data have been compared by Conneely et al. [31] and by
Myerscough and Peach [48] to earlier calculations). Interestingly, the closecoupling calculations of Conneely et al. [31] lie closer to the OPWL results than
they do to the APWL, Robinson-Geltman, or Cooper-Martin results. The
Moskvin PW calculations, not shown in Figure 12, predict a very large crosssection of almost 40 Mb for hv = 0.34 Ry, but the p + s matrix elements are
incorrect [48].
Tables IV and V list u values for Br- and I-, respectively. The energy in these
tables is relative to that for the J = 3/2 level, and is thus a measure of the kinetic
energy only for the process producing that level. The OPW values in these tables
differ from those in Tables 111 and IV by having the intensity distributed over the
two J levels and then summed for a given wavelength. We note that the APWv
values are larger than the APWL values at energies below 0.2 Ry and agree better
with both the OPWL values and the Robinson-Geltman results. As in previous
cases, the OPW, results are lower at their maxima than are the other crosssections, but are larger at energies above 1 Ry. In general, the OPW decrease
more slowly with energy than do the APW cross-sections, reflecting the inadequate treatment of the attractive potential by the former method. For I-, the
830
MOHRAZ AND LOHR
,
I
c160
-
40
-
APW(L)
-P
b
'\
-/-----
20
I
I ,
0.2
0.4
0.6
0.0
PHOTON ENERGY ( R y )
I .o
I.2
Figure 12. Cross-section m in Mb vs. photon energy in Ry for CI-. APW (length)
results with polarization denoted by (- - - -) and OPW (length) results by (- . - .).
Other results are those of Cooper and Martin [27] denoted by (. . . .), Robinson and
and Conneely et al. [311by (- -). Energy shown as photon
Geltman [28] by (-),
energy since all curves include thespin-orbit splitting of 0.008 Ry.
OPWLresults in Figure 8 (without spin-orbit splitting) or Table V (with spin-orbit
splitting) show a slight increase above 6 Ry, although the cross-section decreases
at even higher energies.
In addition to the PW and OPW p values described in Section 3B, Figures 5-8
display APWLP values including core polarization. Values are not shown for
energies below 0.2 Ry as they nearly coincide with the OPW curves. However, the
APWL minima do occur at a lower energy than the PW or OPW minima; for
example, for C1- at approximately 0.07 Ry vs 0.18 Ry. As in the APW study [25]
of inner p-shell photoionization, the dipole length and velocity p values coincide
within 2% or better except in regions where p is varying rapidly with energy, so
the velocity values are not shown. In the energy range from threshold to 3 Ry the
APWL p values tend to be closer to the OPWLand OPWv values than to the PWL
values. However, the second APWL minimum for C1- in Figure 6 occurs near
2.9 Ry rather than above 10 Ry for the second OPWL minimum, probably as a
result of the APW method's more nearly accurate treatment of the s channel. The
APW, curve for CI- passes through zero,at 2.7 and 3.1 Ry, on either side of this
second minima at 2.9 Ry. At 2.7 Ry the d channel matrix element Rd is zero,
while at 3.1 Ry the crossing results from Rd = 2Rs cos Q, in Eq. (21),as does the
lower energy crossing near 0.2 Ry. The APWL p values without polarization are
similar, but with the second minimum shifted to 3.4 Ry. Although APWL p values
CALCULATING PHOTODETACHMENT CROSS-SECHONS
831
TABLEIV. Photodetachment cross-sections (Mb) for Br- with polarization
and spin-orbit coupling.
3674
0.0
3400
0.02
3058
0.05
2866
0.07
12.1
2618
0.1
15.8
2034
0.2
31.0
1663
0.3
45.6
1406
0.4
55.6
1218
0.5
1075
0.6
961
0.7
87 0
0.8
0.0
0.0
0.0
5.1
8.7
10.0
13.8
7.1
15.0
17.6
21.5
16.6
19.4
22.1
17.7
19.7
25.0
30.8
21.1
40.5
39.5
26.7
53.5
44.6
33.6
61.5
59.8
45.9
39.6
64.9
60.3
44.7
43.5
63.6
57.9
41.3
45.8
58.2
54.2
37.2
46.5
50.1
0.0
7 94
0.9
49.0
33.4
45.9
41.1
730
1.0
42.6
29.6
44.8
-
405
2.0
9.5
6.3
24.4
214,
4.0
0.83
1.0
5.9
146
6.0
0.16
0.13
1.2
110
8.0
0.11
0.10
0.8
89
10.0
0.08
0.08
0.8
" E A = 0 . 2 4 8 R y , A E s , = 0 . 0 3 4 Ry.
N O polarization.
Robinson and Geltman, Ref. [28].
€or the other halides were not calculated for energies above 3 Ry, there is no zero
value for Rd €or F- above threshold but below 10 Ry. For Br- and I- there are
zero values of Rd near 7 Ry with polarization and near 8 Ry without, so there must
be at least one crossing in the same energy range as with the OPW results in
Figures 7 and 8."
* Noteadded in proof: Calculations of p by the APW method have now been made for F-, Br-, and
I for electron kinetic energies from 3.0 to 10.0R y in increments of 1.O Ry, thus extending the curves
denoted by (- -) in Figures 5,7, and 8. For F- the curve is nearly flat over this range and approaches a
value of 1.49 at 10.0 Ry in calculations including polarization and using the length operator. For Brthere is minimum near 7.5 Ry, but p is approximately 0.0 instead of - 1.0 as in the OPW, and OPWv
curves. For 1 there is a similar shallow minimum in p near 8.0 Ry. Thus, as already seen for CI- in
Figure 6, the APW calculations do not show the deep minima in p in the range 1.0 to 10.0 Ry that the
OPW calculations show. Both methods d o yield, however, deep minima below 1.0Ry with p
approaching - 1.O. Qualitatively similar results are found in calculations without polarization.
~
832
MOHRAZ AND LOHR
4. summary
From the results and discussion in the previous section some conclusions may
be drawn about the use of OPW continuum functions in describing photodetachment and photoionization processes:
1. The operator which should be used in connection with OPW continuum
functions is the dipole length operator. The dipole velocity operator is not suitable
in this case since the PW is an eigenfunction of this operator. A comparison of the
TABLEV. Photodetachment cross-sections (Mb) for 1- with polarization and
spin-orbit coupling.
h(i)a
E (Ry)
oL(APW)
0.0
4032
0.0
3861
0.01
1.8
3302
0.05
3.9
'"(APW)
5L(OPW)b
'L(R~G)'
0.0
0.0
9.3
10.5
16.6
13.8
15.5
17.5
0.0
3079
0.07
5.5
15.9
16.2
27.0
2978
0.08
7.6
20.6
21.4
30.3
2795
0.1
10.9
24.6
25.2
37.2
213 9
0.2
32.2
41.4
28.6
69.0
1732
0.3
59.0
54.8
38.4
92.8
1456
0.4
82.0
60.8
51.1
97.7
1255
0.5
91.8
59.5
62.7
89.0
1103
0.6
91.4
53.0
70.4
73.3
984
0.7
83.4
46.0
73.8
56.4
888
0.8
70.7
38.9
73.1
41.6
809
0.9
59.0
32.1
70.1
30.2
743
1.0
48.2
25.7
62.0
409
2.0
6.4
4.2
19.3
216
4.0
0.78
0.49
1.6
146
6.0
0.20
0.15
1.2
111
8.0
0.10
0.09
3.1
0.08
0.07
3.5
89
10.0
-
EA = 0.226 Ry, AE,, = 0.69 Ry.
NO polarization.
'Robinson and Geltman, Ref. [28].
a
OPWLand OPWv cross-sections with the corresponding APW values, with other
calculated values, and with experimental values, shows that there is a better
agreement using the length operator. In addition, as pointed out by Reed et al.
CALCULATING PHOTODETACHMENT CROSS-SECTIONS
833
[19], only the length operator gives the correct threshold behavior for p-electron
detachment into a PW final state. Most of the published PW and OPW results
[I-1 13for the photoionization of neutral molecules have, however, for reasons of
convenience, been obtained exclusively with the velocity operator.
2. In OPW and PW methods the continuum states are essentially free waves.
As such, they are better approximations for those photodetachment processes
which result in continuum states with small phase shifts, namely states with high
angular momentum. Therefore, one would expect that PW and OPW functions
are poor for describing photodetachment from a p-orbital where one of the final
states is an s wave. As has also been frequently pointed out, these functions are
better approximations to continuum states at high rather than low energies.
3. Although the OPWLvalues of both (T and p as a function of photon energy
are in some instances in reasonable agreement with more reliable values, there is
sufficient disagreement, even for atomic photodetachment, to suggest that indiscriminate use of the OPW method to describe photoionization of neutrals is
unwarranted. In particular, we share the recent concern [37] about the significance of OPW p values [lo] for molecular photoionization.
It is obvious that the PW and OPW methods are far less accurate (at least at
very low electron energies) than the APW and other theoretical methods discussed here, as they d o not correctly account for the interaction between the ejected
electron and the atomic core. However, the main attraction of the method is that it
can be readily extended to molecular systems for which the other method are not
easily applicable. Computationally, it is by far the cheapest method for crosssection calculations as all the integrals can be evaluated analytically.
The major advantage of the APW method* over the other related theoretical
methods is that the APW method offers a more convenient and computationally
cheaper means of obtaining continuum wave-functions. Our formulation of the
method, however, does not explicitly include polarization and exchange potentials. In all of the other methods reviewed here, at least the continuum function
and sometimes both the bound and continuum functions, are obtained by the
numcrical integration of the corresponding radial wave equation. In the APW
method, however, the solutions of the radial equation for each region of the
potential are known. The continuum-state solution for the entire range of
potential is constructed from these “regional” solutions, smoothly joined together
by the application of suitable boundary conditions.
Acknowledgment
The authors wish to thank the University of Michigan Computing Center for
the use of its facilities. One of us (L.L.L.) would like to thank the Department of
Chemistry of the University of California, Berkeley, for its hospitality during the
writing of this paper.
* The term APW is usually used for a related method for solids involving numerical integrations
within a spherical reaon and including exchange via a statistical approximation [49].
834
MOHRAZ A N D LOHR
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Received August 6, 1975